The concept of a function

Observe the following relationships between quantities and determine, in each case, whether it is a function or not:

1. a) Each mobile phone number is associated with the name of its owner in the contact list.
2. b) Each student in your class is associated with their height measured in centimetres.
3. c) Each supermarket barcode is associated with the price of the corresponding product.
4. d) Each person is associated with the eye colour of their siblings.

At a secondary school, different relationships between quantities are being analysed. Determine, giving reasons, which of the following situations correspond to a function and, if so, identify the independent variable and the dependent variable:

1. a) Each student's ID number is associated with their current age.
2. b) Each maximum temperature reached in the playground during a day is associated with the times at which that temperature was recorded.
3. c) The amount of gigabytes ($GB$) consumed on a mobile tariff is associated with the monthly bill price.
4. d) Each natural number is associated with its factors.

In an app store, each premium game has a fixed price of $2.50$ €. We want to analyse the relationship between the number of games purchased and the total cost of the purchase.

1. a) Identify which is the independent variable and which is the dependent variable.
2. b) Is this relationship a function? Give reasons for your answer based on the definition of a function.
3. c) Write the algebraic expression (formula) that relates both quantities, letting $x$ be the number of games and $y$ be the total cost.

Given the function defined by the algebraic expression $f(x) = 3x - 2$, complete the following table of values in your notebook:

$x$ (Input)	$-2$	$-1$	$0$	$1$	$2$
$f(x)$ (Output)

Once completed, state the image of $x = -1$.

The price of a monthly subscription to a streaming service is $9$ €, but there is a special offer for the first month and it only costs $3$ €.

1. a) Draw up a table relating the number of subscribed months (from $1$ to $4$ months) to the total cumulative expenditure.
2. b) Plot these four points on a Cartesian coordinate system.
3. c) Does it make sense to connect the points with a continuous line? Explain why.

Given the function $f(x) = \frac{x^2 - 4}{2}$, complete the following tasks:

1. a) Calculate the images of the values $x = 0$, $x = 2$ and $x = -4$.
2. b) Determine the value of $x$ (pre-image) whose image is $f(x) = 6$.
3. c) Does the point $P(-1; -1.5)$ belong to the graph of the function? Justify your answer by showing the necessary calculations.

For a graph to represent a function, any vertical line we draw must cut the graph at a single point. Use this "vertical line test" to decide which of the following graphs is NOT a function:

Justify your choice by explaining what happens on the chosen graph when you draw a vertical line.

Observe the following graphs plotted on the Cartesian plane. Using the vertical line test, state with reasons which of them represents a function and which does not.

Analyse the following graphs and determine whether or not they represent a function. Justify your answer using the concept of the relationship between the independent and dependent variables.

1. a) $x^2 + y^2 = 25$ (A circle of radius 5 centred at the origin).
2. b) $y = \sqrt{x}$ (Considering only the positive root).
3. c) A vertical line on the Cartesian plane passing through $x = 4$.
4. d) A sideways parabola whose equation is $x = y^2$.

Aoife is monitoring her smartphone's battery consumption while playing a video game. The following graph shows the percentage of battery remaining according to the time spent playing (in minutes):

From the graph, answer the following:

1. a) What percentage of battery was there when she started playing (time = $0$)?
2. b) How much battery remains after $20$ minutes of playing?
3. c) If the battery reaches $0\%$, how many minutes has she been playing in total?

Observe the points plotted on the following Cartesian plane, which show the temperature of a greenhouse during the early hours of the morning:

State the coordinates $(x, y)$ of the four points marked on the graph and explain what each of them means in the context of the problem. (Example: At hour $1$, the temperature is...).

A hiker records the percentage of battery remaining on their smartphone throughout a mountain walk. The resulting graph is as follows:

From the graph, answer the following:

1. a) Which are the independent and dependent variables? State the units of measurement.
2. b) What was the battery percentage at the start of the hike?
3. c) Is it an increasing or decreasing function? What is the physical meaning of this in the context of the problem?
4. d) Estimate how much time has passed when the phone has exactly $50$% charge remaining.

The following table shows the amount of water (in litres) remaining in a paddling pool while it is being emptied using an extraction pump:

Time (minutes)	0	5	10	15	20
Water (litres)	4500	3600	2700	1800	900

1. a) Is it a functional relationship? Explain why.
2. b) Plot the points on a coordinate system using a suitable scale.
3. c) If the trend remains constant, how long will it take for the pool to empty completely?
4. d) What volume of water will remain $12$ minutes after emptying begins?

You can do it on paper and send a photo with your answer.

Consider the relation that associates each integer $x$ with its modulus plus three units, that is, $y = |x| + 3$.

1. a) Draw up a table of values for the integers between $-3$ and $3$.
2. b) Plot the points obtained on a Cartesian coordinate system.
3. c) Does it make sense to connect the points with a continuous line if the problem states that $x$ is an integer? Explain why.
4. d) State the range (set of values taken by the dependent variable) based on your table and graph.

You can do it on paper and send a photo with your answer.

A video game programmer is designing the reward system for a new level. The amount of "Stellar Credits" ($C$) that a player receives depends on the time ($t$) in minutes taken to complete a mission, according to the relationship:

$$C(t) = \frac{600}{t-5}$$

The figure shows the graph of this relationship for the times the game allows.

1. a) State with reasoning whether this relationship is a function for any value of $t$ greater than zero.

1. b) If the minimum time to physically complete the mission is 6 minutes and the maximum allowed by the server is 60 minutes, determine the domain and the range of this function in this real-life context.

1. c) What happens to the reward as the player gets closer to the time of 5 minutes? Would it make sense for the domain to include the value $t = 5$?

A hiker climbs a mountain following a trail. The function that relates their altitude above sea level, $H$ (in metres), to the horizontal distance covered, $d$ (in kilometres), is $H(d) = 400 + 150d$.

1. a) At what altitude is the starting point of the route ($d = 0$)?
2. b) If the hiker has covered $3$ km horizontally, at what altitude are they?
3. c) Identify the variables of the function and explain their physical meaning.

An e-scooter rental company charges an unlock fee of $1.20$ € and a variable rate of $0.15$ € for each minute of use.

1. a) Identify the two variables involved and classify them as dependent and independent.
2. b) Write the algebraic expression or formula that allows you to calculate the total rental cost, $y$, in terms of the time used in minutes, $x$.
3. c) Calculate how much a student will pay if they use the scooter for $22$ minutes to travel from home to school.

The area of a rectangle whose base is twice its height can be expressed as a function.

1. a) If $x$ represents the height of the rectangle, write the algebraic expression for the area function $A(x)$.
2. b) Which is the independent variable and which is the dependent variable?
3. c) Calculate the area when the height is $4.5$ $cm$.
4. d) If the area of the rectangle is $50$ $cm^2$, what is its height?

A rectangular wooden frame is to be built for a mirror. We have a wooden batten with a total length of $4$ metres to make the entire perimeter.

1. a) If $x$ is the length of the base of the frame (in metres), express the function $A(x)$ that calculates the area enclosed by the frame.
2. b) What is the domain of this function taking into account the physical constraints of the problem?
3. c) Calculate the area when the base measures $0.5 m$ and when it measures $1.5 m$.
4. d) For what value of $x$ is the maximum area obtained? What geometric shape does the frame have in that case?

A water tank has the shape of an inverted cone. It is being filled with water at a constant rate. The relationship between the volume of water ($V$) and the height reached by the water ($h$) is given by the function:

$$V(h) = \frac{1}{3} \times \pi \times (0.5h)^2 \times h$$

1. a) Simplify the expression for the function $V(h)$.
2. b) Is it a linear function? Justify your answer by observing the degree of the variable $h$.
3. c) Calculate the pre-image of $V = \frac{2\pi}{3}$, that is, the height $h$ required to have that volume.
4. d) If the tank has a total height of $4 metros$, what are the domain and range of the function in this situation?

Observe the following table of values relating two quantities, $A$ and $B$:

Quantity A ($x$)	-2	-1	0	1	2
Quantity B ($y$)	5	2	1	2	5

1. a) Find an algebraic expression (formula) relating $y$ to $x$.
2. b) State whether it is a function and justify why.
3. c) Determine the axis intercepts.
4. d) If we extend the table with the value $x = 1$ and assign it $y = 3$, would it still be a function? Give reasons for your answer.

A drone delivery company charges for its services as follows: a fixed cost of $\text{€}\ 5$ per booking and, in addition, $\text{€}\ 2$ for each kilogram of weight. However, if the package exceeds $10\,\text{kg}$, the cost per additional kilogram (only the kilograms beyond 10) is halved.

1. a) Write the function $P(x)$ that represents the total price of the delivery in terms of the kilograms $x$ of the package. Note: the function must have two expressions (a piecewise function).

1. b) Calculate $P(8)$ and $P(15)$.

1. c) Graph the function.

1. d) If a customer has paid $\text{€}\ 31$, how many kilograms did their package weigh?

You can do it on paper and send a photo with your answer.

A telephone company offers a teenage plan called "Gamer Pro". The monthly cost ($C$) in euros depends on the Gigabytes ($g$) used:

• The first $20\ \text{GB}$ are free (included in the fixed fee of $\text{€}\ 15$).

• Beyond $20\ \text{GB}$, each extra Gigabyte costs $\text{€}\ 1.5$, but only up to a maximum of $50\ \text{GB}$ of total usage.

• If more than $50\ \text{GB}$ are used, the speed is reduced and nothing more is charged.

1. a) Graph the function $C(g)$ for a usage between $0$ and $80\ \text{GB}$.

1. b) Define the function by means of a piecewise analytical expression.

1. c) State the intervals where the function is constant and where it is increasing.

1. d) What are the domain and the range of this function?

You can do it on paper and send a photo with your answer.

Consider the relation that associates each real number $x$ with its distance to the nearest integer. For example, if $x = 2.3$, its distance to the nearest integer (2) is $0.3$. If $x = 2.8$, its distance to the nearest integer (3) is $0.2$.

1. a) Plot this relation graphically on the interval $x \in [0, 3]$.
2. b) Explain why this relation is a function.
3. c) Determine the range of the function.
4. d) Is it a periodic function? If so, state its period.

You can do it on paper and send a photo with your answer.